Convergence Rate of ELEGANT Algorithms Jan de Leeuw Version 19, December 02, 2016 Abstract We study the convergence rate of the ELEGANT algorithm for squared distance scaling by using both observed convergence rates and an analytical expression for the derivative of the algorithmic map. Contents 1 Introduction 1 2 Derivatives of the ELEGANT transform 3 2.1 Derivatives of Gamma .............................. 3 2.2 Derivatives of B .................................. 4 2.3 Chain Rule .................................... 5 3 Computing Convergence Rates 5 3.1 Small Example .................................. 6 3.2 Ekman Example ................................. 7 4 Appendix: Code 8 4.1 auxilary.R ..................................... 8 4.2 elegant.R ..................................... 11 4.3 partials.R ..................................... 13 References 16 Note: This is a working paper which will be expanded/updated frequently. All suggestions for improvement are welcome. The directory gifi.stat.ucla.edu/speed has a pdf version, the complete Rmd file with all code chunks, the bib file, and the R source code. 1 Introduction The multidimensional scaling (MDS) loss function sstress is defined on the set R n×p of n × p configuration matrices as σ(X ) := 1 2 n  j =1 n  j =1 w ij (δ 2 ij − d 2 ij (X )) 2 , (1) 1